Intercalation of molecular species provides a route to tailor the electronic structure in van der Waals superconductors, yet intercalated compounds have rarely been integrated into phase-sensitive quantum devices. Here, we realize hybrid molecular Josephson junctions based on TaS2/NbSe2 heterostructures. Selective intercalation of the TaS2 electrode induces monolayer-like Ising superconductivity, while the junctions exhibit asymmetric Fraunhofer interference, field-induced and spontaneous Josephson diode effects. The zero-field diode disappears upon reversing the electrode stacking and re-emerges after intercalating the lower NbSe2 electrode. These findings indicate that the hybrid superconducting electrode hosts unconventional superconductivity underlying the nonreciprocal transport, while molecular intercalation acts as a tuning knob for its internal phase structure. This behavior is consistent with a time-reversal-symmetry-breaking multiband superconducting state characterized by a nontrivial relative phase between superconducting condensates. Our results establish hybrid molecular Josephson junctions as a quantum-device platform for phase-sensitive investigation and integration of molecularly engineered superconductors, opening opportunities to uncover and engineer unconventional superconducting phases.